2 Preliminaries
2.1 Pairs of 2-forms
Throughout this section, let \(V\) denote an oriented \(4\)-dimensional real vector space. Fix a volume form \(\varepsilon \in \Lambda^{4}V^*\) which induces the given orientation. Given two \(2\)-forms \(\omega,\phi \in \Lambda^{2}V^*\), we may write \(\omega\wedge\phi=\langle \omega,\phi \rangle \varepsilon\) for some unique real number \(\langle \omega,\phi \rangle.\) Clearly the map \((\omega,\phi) \mapsto \langle \omega,\phi\rangle\) defines a symmetric bilinear form on the \(6\)-dimensional real vector space \(\Lambda^{2}V^*\) which is easily seen to be nondegenerate and of signature \((3,3).\) Replacing \(\varepsilon\) with another orientation compatible volume form gives a bilinear form which is a positive multiple of \(\langle\cdot\,,\cdot\rangle.\) Consequently, the wedge product may be thought of as a conformal structure of split signature on \(\Lambda^2V^*.\)
A pair of \(2\)-forms \(\omega,\phi \in \Lambda^2V^*\) is called elliptic if \[\langle \omega,\omega \rangle\langle \phi,\phi \rangle>\langle \omega,\phi \rangle^2.\]
It is a natural problem to classify the pairs of elliptic \(2\)-forms on \(V.\) This is a special case of a more general problem: Let \(\omega \in \Lambda^2V^*\) be a symplectic \(2\)-form whose stabiliser subgroup will be denoted by \(\mathrm{Sp}(\omega) \subset \mathrm{GL}(V).\) The natural representation of \(\mathrm{Sp}(\omega)\) on \(\Lambda^2V^*\) decomposes as \(\Lambda^2V^*=\left\{\omega\right\} \oplus \omega^{\perp}\) where both summands are irreducible \(\mathrm{Sp}(\omega)\)-modules. Here \(\omega^{\perp}\) is the \(5\)-dimensional linear subspace of \(\Lambda^2V^*\) consisting of \(2\)-forms orthogonal to \(\omega.\) One can ask to classify the orbits of \(\mathrm{Sp}(\omega)\) on \(\omega^{\perp}.\) This has been carried out in [8] and in the elliptic case one obtains:
Let \(\omega,\phi \in \Lambda^2V^*\) be a pair of elliptic orthogonal \(2\)-forms, then there exists a positive real number \(\kappa\) and a basis \(e^i\) of \(V^*\) such that \[\omega=e^1\wedge e^3-e^2 \wedge e^4, \quad \phi=\kappa\left(e^1\wedge e^4+e^2\wedge e^3\right).\]
The constant \(\kappa\) is an \(\mathrm{Sp}(\omega)\)-invariant and thus parametrises the set of elliptic \(\mathrm{Sp}(\omega)\)-orbits. Ellipticity will be useful because of the following:
Let \(W\) be \(3\)-dimensional real vector space. Then the pullback of an elliptic pair of \(2\)-forms \(\omega,\phi \in \Lambda^2V^*\) with any injective linear map \(A : W \to V\) gives two linearly independent \(2\)-forms on \(W.\)
Proof. The ellipticity condition is equivalent to every nonzero linear combination of \((\omega,\phi)\) being symplectic. Suppose \((\omega,\phi)\) is an elliptic pair of \(2\)-forms. Then for every choice of real numbers \((\lambda_1,\lambda_2)\neq 0\), the \(2\)-form \(\tau=\lambda_1\omega+\lambda_2\phi\) is symplectic. Since there are no isotropic subspaces of dimension greater than \(2\) in the symplectic vector space \((V,\tau)\), it follows that \(A^*\tau=\lambda_1A^*\omega+\lambda_2A^*\phi\neq 0\) for every linear injective map \(A : W \to V.\)
2.2 Splittings of complex structures
Let \(\mathcal{C}^+(V)\) denote space of complex structures on \(V\) which are compatible with the orientation, i.e. its points \(J\in \operatorname{End}(V)\) satisfy \(\varepsilon(v_1,Jv_1,v_2,Jv_2)\geq 0\) for all vectors \(v_1,v_2 \in V.\) Moreover let \(G^+_2(\Lambda^2V^*,\wedge_+)\) denote the submanifold of the Grassmannian of oriented \(2\)-planes in \(\Lambda^2V^*\) to whose elements the wedge product restricts to be positive definite. Given a \((2,\! 0)\)-form \(\alpha \in \Lambda^{2,0}V^*\) with respect to some \(J \in \mathcal{C}^+(V)\), let \(\Lambda_J \in G^+_2(\Lambda^2V^*,\wedge_+)\) denote the \(2\)-dimensional linear subspace spanned by \(\operatorname{Re}(\alpha)\), \(\operatorname{Im}(\alpha)\) and orient \(\Lambda_J\) by declaring \(\operatorname{Re}(\alpha),\operatorname{Im}(\alpha)\) to be positively oriented. Clearly \(\Lambda_J\) and its orientation are independent of the chosen \((2,\! 0)\)-form \(\alpha\) and one thus obtains a map \(\psi : \mathcal{C}^+(V) \to G^+_2(\Lambda^2V^*,\wedge_+)\) given by \(J \mapsto \Lambda_{J}.\) Note that \(G=\mathrm{GL}^+(V)\) acts smoothly and transitively from the left on \(\mathcal{C}^+(V)\) via \((A,J) \mapsto A^{-1}JA.\) Every element of \(G^+_2(\Lambda^2V^*,\wedge_+)\) admits a positively oriented elliptic conformal basis. It follows with Lemma 2.2 that via pushforward, \(\mathrm{GL}^+(V)\) acts smoothly and transitively from the left on \(G^+_2(\Lambda^2V^*,\wedge_+)\) as well.
The map \(\psi : \mathcal{C}^+(V) \to G^+_2(\Lambda^2V^*,\wedge_+)\), \(J \mapsto \Lambda_J\) is a \(G\)-equivariant diffeomorphism.
Proof. Clearly the map \(\psi\) is \(G\)-equivariant. To prove that \(\psi\) is a diffeomorphism it is sufficient to show that \(G_J=G_{\psi(J)}\) for all \(J \in \mathcal{C}^+(V)\) where \(G_J\) and \(G_{\psi(J)}\) denote the stabiliser subgroups of \(G\) with respect to \(J\) and \(\psi(J)\) respectively. Choose \(J \in \mathcal{C}^+(V)\), then we have \(G_J \subset G_{\psi(J)}.\) Write \[J(v)=-e^2(v)e_1+e^1(v)e_2-e^4(v)e_3+e^3(v)e_4\] for some basis \((e_i)\) of \(V\) and dual basis \((e^i)\) of \(V^*.\) Then \[\omega=e^1\wedge e^3-e^2\wedge e^4=\frac{1}{2}w_{kl}e^k\wedge e^l, \quad \phi=e^1\wedge e^4+e^2\wedge e^3=\frac{1}{2}f_{kl}e^k\wedge e^l\] is a positively oriented conformal basis of \(\Lambda_J.\) Consequently every \(A \in G_{\psi(J)}\) satisfies \(A^*\omega=x\omega+y\phi\) and \(A^*\phi=-y\omega+x\phi,\) for some real numbers \((x,y)\neq 0.\) The matrix representation \(a\) of \(A\) with respect to the basis \((e_i)\) thus satisfies \[a^twa=xw+yf, \quad a^tfa=-yw+xf.\] From this one easily concludes \(awf=wfa\) which is equivalent to \(A\) commuting with \(J.\)
Proposition 2.4 motivates the following:
A splitting of a complex structure \(J\) on \(V\) is a pair of lines \(L_1,L_2 \in \mathbb{P}(\Lambda^2V^*)\) such that \(\Lambda_{J}=L_1\oplus L_2.\)
Call two \(4\)-dimensional real vector spaces \(V\), \(V^{\prime}\) equipped with complex structures \(J\), \(J^{\prime}\) and splittings \((L_1,L_2)\), \((L_1^{\prime},L_2^{\prime})\) equivalent, if there exists a complex linear map \(A : V \to V^{\prime}\) such that \(A^*(L_i^{\prime})=L_i\) for \(i=1,2.\)
On \(V=\mathbb{R}^4\) let \(\omega_0=e^1\wedge e^3-e^2\wedge e^4\) and \(\phi_0=e^1\wedge e^4+e^2\wedge e^3\) where \(e^1,\ldots,e^4\) denotes the standard basis of \((\mathbb{R}^4)^*.\) Define \(L_1=\left\{\omega_0\right\}\) and \(L_2=\left\{\alpha \omega_0+\phi_0\right\}\) for some nonnegative real number \(\alpha.\) Orient \(L_1\oplus L_2\) by declaring \(\omega_0,\phi_0\) to be a positively oriented basis and let \(J_0\) be the associated complex structure. Then \(S_\alpha=(L_1,L_2)\) is a splitting of \(J_0.\)
Every pair \((V,J)\) equipped with a splitting \((L_1,L_2)\) is equivalent to \((\mathbb{R}^4,J_0)\) equipped with the splitting \(S_{\alpha}\) for some unique \(\alpha \in \mathbb{R}^+_0.\)
Proof. Let \(L_1=\{\omega\}\) and \(L_2=\{\omega^{\prime}\}\) for some \(2\)-forms \(\omega,\omega^{\prime} \in \Lambda^2 V^*.\) Since the wedge product restricts to be positive definite on \(L_1\oplus L_2\) we have \(\omega\wedge\omega >0\) and there exists a real number \(\alpha\), such that \(\omega^{\prime}=\alpha\omega+\phi\) for some \(2\)-form \(\phi\) satisfying \(\omega\wedge \phi=0\) and \(\phi \wedge \phi>0.\) After possibly rescaling \(\omega^{\prime}\) we can assume that \(\phi\wedge\phi=\omega\wedge\omega\) and that \(\alpha\) is nonnegative. It follows with Lemma 2.2 that there exists a linear map \(A : V \to \mathbb{R}^4\) which identifies \(\omega\) with \(\omega_0\) and \(\phi\) with \(\phi_0\), in particular \(A\) is complex linear. To prove uniqueness of \(\alpha\) suppose \(A : \mathbb{R}^4 \to \mathbb{R}^4\) satisfies \(A^*\omega_0=x\omega_0\) and \(A^*(\alpha \omega_0+\phi_0)=y\left(\beta\omega_0+\phi_0\right)\) for some real numbers \(x,y\neq 0\) and some nonnegative real numbers \(\alpha,\beta.\) Then \(A^*\left(\omega_0\wedge\omega_0\right)=x^2\omega_0\wedge\omega_0\) and consequently \[A^*\left(\omega_0\wedge(\alpha\omega_0+\phi_0)\right)=\alpha x^2\omega_0\wedge\omega_0=xy\beta\omega_0\wedge\omega_0,\] which is equivalent to \(\alpha x = \beta y.\) We also have \[A^*\left((\alpha \omega_0+\phi_0)\wedge (\alpha \omega_0+\phi_0)\right)=x^2(\alpha^2+1)\omega_0\wedge\omega_0=y^2(\beta^2+1)\omega_0\wedge\omega_0,\] which implies \(x^2=y^2\) and thus \(\alpha^2=\beta^2.\) Since \(\alpha,\beta \geq 0\), the claim follows.
For a splitting \((L_1,L_2)\), the unique nonnegative real number \(\alpha\) provided by Proposition 2.6 will be called the degree of the splitting. A splitting of degree \(0\) will be called orthogonal.
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